where [−][-] denotes a bracket type. Since every type maps to its bracket, Decidable1(A)Decidable1(A) implies Decidable2(A)Decidable2(A).

On the other hand, if Decidable2(A)Decidable2(A) holds and AA is an h-set, i.e. it satisfies uniqueness of identity proofs, then (x=y)(x=y) and ¬(x=y)\neg (x=y) represent disjoint subobjects of A×AA\times A. Thus (x=y)+¬(x=y)(x=y) + \neg (x=y) is already a subobject of A×AA\times A, so it is equivalent to its bracket, and Decidable1(A)Decidable1(A) also holds.

The converse of this is also true: if Decidable1(A)Decidable1(A) holds, then not only does Decidable2(A)Decidable2(A) also hold, but in fact AA is an h-set. This was first proven by Michael Hedberg; a proof can be found at h-set and in the references below. This fact is useful in homotopy type theory to show that many familiar types, such as the natural numbers, are h-sets.

For non-h-sets, the difference between Decidable1Decidable1 and Decidable2Decidable2 can be dramatic. For instance, if we model homotopy type theory in a Boolean(∞,1)(\infty,1)-topos (such as ∞Gpd\infty Gpd constructed classically), then every type satisfies Decidable2Decidable2 (which is what it means for the logic to be boolean), but only the h-sets satisfy Decidable1Decidable1 (in accordance with Hedberg's theorem).